Complete Reduction for Derivatives in Transcendental Liouvillian Extensions

A complete reduction ϕ for derivatives in a differential field F is a linear idempotent on F over its constant subfield whose kernel is equal to the subspace consisting of derivatives in F. It enables us to decompose an element f as the sum of a derivative and ϕ ( f) such that f is a derivative in F if and only if ϕ ( f) = 0.

We outline a complete reduction algorithmically for derivatives in a transcendental Liouvillian extension of the field of rational functions. Typical examples for transcendental Liouvillian extensions are differential fields generated by (poly-)logarithmic functions, hyperexponential functions, as well as the logarithmic integral, the exponential integral and the error function. Such extensions may contain non-elementary and non-D-finite functions.